On 05 Nov 2011, at 17:22, Jason Resch wrote:

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On Fri, Nov 4, 2011 at 10:35 AM, Bruno Marchal <marc...@ulb.ac.be>wrote:- the Mandelbrot set + zooms (I suspect) etc. Bruno,This idea intrigued me. Could you provide some elaboration on howzooms of the Mandelbrot set could be universal?

`I am talking of the rational Mandelbrot set. I conjecture that the`

`universal question "x belongs to W_y" can be reduced to the question a/`

`b + c/d i belongs to the rational Mandelbrot set, with a, b, c, d`

`being integers. (W_y = the domain of the yth partial computable`

`function phi_y)`

`This wouldn't be as astonishing than reducing "x belongs to W_y" to`

`the question "are x and y solutions to some degree 4 polynomial`

`diophantine equation (and the possibility of this follows from`

`Matiyazevitch theorem).`

`If a rational complex number belongs to the M set, by zooming long`

`enough on it, you will know if it is the case, but if it does not`

`belong, it might be that you will never know it.`

`In the case of the notion of computability "on a ring" (like R or C)`

`Blum, Shub and Smale(*) have solved that conjecture positively, but`

`this cannot be used to make the rational M set universal. I think that`

`it is still an open problem.`

`Technically, I have a beginning of a proof based on the idea that the`

`Mandelbrot set (well, its complementary) can be seen as the set of z`

`such that z does not belong to its Julia set, making the Mandelbrot`

`set diagonalizing on the Julia sets. Unlike Julia sets it inherits`

`some closure property, which makes me finding such idea (the`

`universality of the M set) rather plausible.`

`That would be nice, because it would make the M set a compact view of`

`UD* (a complete execution of a universal dovetailer).`

`Each little Mandelbrot sets are surrounded by infinite bifurcation`

`processes which have the little M set as limit, that can illustrate`

`the complexity of the comp measure problem, because such a limit is`

`"geometrically" infinitely complex, yet algorithmically very simple`

`(as the code for the M set is very short).`

Note that Penrose has made a similar conjecture. (*) http://www.amazon.com/Complexity-Real-Computation-Lenore-Blum/dp/0387982817 Bruno http://iridia.ulb.ac.be/~marchal/ -- You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.